Optimal. Leaf size=191 \[ \frac{12 b^2 n^2 x \cos ^2\left (a+b \log \left (c x^n\right )\right )}{64 b^4 n^4+20 b^2 n^2+1}+\frac{x \cos ^4\left (a+b \log \left (c x^n\right )\right )}{16 b^2 n^2+1}+\frac{4 b n x \sin \left (a+b \log \left (c x^n\right )\right ) \cos ^3\left (a+b \log \left (c x^n\right )\right )}{16 b^2 n^2+1}+\frac{24 b^3 n^3 x \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{64 b^4 n^4+20 b^2 n^2+1}+\frac{24 b^4 n^4 x}{64 b^4 n^4+20 b^2 n^2+1} \]
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Rubi [A] time = 0.0449368, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {4478, 8} \[ \frac{12 b^2 n^2 x \cos ^2\left (a+b \log \left (c x^n\right )\right )}{64 b^4 n^4+20 b^2 n^2+1}+\frac{x \cos ^4\left (a+b \log \left (c x^n\right )\right )}{16 b^2 n^2+1}+\frac{4 b n x \sin \left (a+b \log \left (c x^n\right )\right ) \cos ^3\left (a+b \log \left (c x^n\right )\right )}{16 b^2 n^2+1}+\frac{24 b^3 n^3 x \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{64 b^4 n^4+20 b^2 n^2+1}+\frac{24 b^4 n^4 x}{64 b^4 n^4+20 b^2 n^2+1} \]
Antiderivative was successfully verified.
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Rule 4478
Rule 8
Rubi steps
\begin{align*} \int \cos ^4\left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{x \cos ^4\left (a+b \log \left (c x^n\right )\right )}{1+16 b^2 n^2}+\frac{4 b n x \cos ^3\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{1+16 b^2 n^2}+\frac{\left (12 b^2 n^2\right ) \int \cos ^2\left (a+b \log \left (c x^n\right )\right ) \, dx}{1+16 b^2 n^2}\\ &=\frac{12 b^2 n^2 x \cos ^2\left (a+b \log \left (c x^n\right )\right )}{1+20 b^2 n^2+64 b^4 n^4}+\frac{x \cos ^4\left (a+b \log \left (c x^n\right )\right )}{1+16 b^2 n^2}+\frac{24 b^3 n^3 x \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{1+20 b^2 n^2+64 b^4 n^4}+\frac{4 b n x \cos ^3\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{1+16 b^2 n^2}+\frac{\left (24 b^4 n^4\right ) \int 1 \, dx}{1+20 b^2 n^2+64 b^4 n^4}\\ &=\frac{24 b^4 n^4 x}{1+20 b^2 n^2+64 b^4 n^4}+\frac{12 b^2 n^2 x \cos ^2\left (a+b \log \left (c x^n\right )\right )}{1+20 b^2 n^2+64 b^4 n^4}+\frac{x \cos ^4\left (a+b \log \left (c x^n\right )\right )}{1+16 b^2 n^2}+\frac{24 b^3 n^3 x \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{1+20 b^2 n^2+64 b^4 n^4}+\frac{4 b n x \cos ^3\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{1+16 b^2 n^2}\\ \end{align*}
Mathematica [A] time = 0.434809, size = 167, normalized size = 0.87 \[ \frac{x \left (128 b^3 n^3 \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+16 b^3 n^3 \sin \left (4 \left (a+b \log \left (c x^n\right )\right )\right )+\left (64 b^2 n^2+4\right ) \cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+\left (4 b^2 n^2+1\right ) \cos \left (4 \left (a+b \log \left (c x^n\right )\right )\right )+8 b n \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+4 b n \sin \left (4 \left (a+b \log \left (c x^n\right )\right )\right )+192 b^4 n^4+60 b^2 n^2+3\right )}{8 \left (64 b^4 n^4+20 b^2 n^2+1\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.09, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{4}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.27288, size = 1455, normalized size = 7.62 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.514719, size = 381, normalized size = 1.99 \begin{align*} \frac{24 \, b^{4} n^{4} x + 12 \, b^{2} n^{2} x \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} +{\left (4 \, b^{2} n^{2} + 1\right )} x \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 4 \,{\left (6 \, b^{3} n^{3} x \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) +{\left (4 \, b^{3} n^{3} + b n\right )} x \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3}\right )} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{64 \, b^{4} n^{4} + 20 \, b^{2} n^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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